📚 Exercise 3F: Graph Transformations | 练习3F：图像变换
In IB Mathematics, Exercise 3F typically focuses on applying transformations to the graphs of functions. Whether you are working with quadratic, reciprocal, or trigonometric functions, mastering translations, stretches, and reflections is essential. This article breaks down the key techniques needed to tackle Exercise 3F with confidence, providing clear methods for each type of transformation and how to combine them effectively.
在IB数学中，练习3F通常聚焦于对函数图像进行变换。无论你面对的是二次函数、倒数函数还是三角函数，掌握平移、伸缩和对称反射都是必备技能。本文将拆解攻克练习3F所需的关键技巧，清晰地讲解每种变换的方法以及如何有效组合它们。
1. Understanding Function Notation | 理解函数符号
ble with function notation y = f(x). Any change inside the argument f(x + a) affects the x‑coordinates, while changes outside, such as f(x) + b, affect the y‑coordinates. This distinction is the foundation of all graph transformations.
在应用变换之前，你必须熟练掌握函数符号 y = f(x)。括号内自变量的任何变化，如 f(x + a)，会影响 x 坐标；而括号外的变化，如 f(x) + b，则影响 y 坐标。这一区别是所有图像变换的基础。
2. Vertical Translations | 垂直平移
f y = f(x) is transformed to y = f(x) + d, the graph moves d units upwards when d > 0, and |d| downwards when d < 0. For example, y = x² + 3 is the parabola y = x² moved up by 3 units.
垂直平移将图像向上或向下移动。若 y = f(x) 变为 y = f(x) + d，则当 d > 0 时图像上移 d 个单位，d < 0 时下移 |d| 个单位。例如，y = x² + 3 就是将抛物线 y = x² 向上平移3个单位得到的。
3. Horizontal Translations | 水平平移
ranslations shift the graph left or right. The transformation y = f(x + c) moves the graph –c units horizontally. If c is negative, say f(x – 2), the graph moves 2 units to the right. This often feels counter‑ remember: replacing x with (x + c) moves the graph in the opposite direction of the sign.
水平平移将图像向左或向右移动。变换 y = f(x + c) 会将图像水平移动 –c 个单位。如果 c 为负，比如 f(x – 2)，图像则向右移动2个单位。这常常有悖直觉，因此请记住：将 x 替换为 (x + c) 会使图像朝与符号相反的方向移动。
4. Reflections in the Axes | 坐标轴对称反射
Reflection in the x‑axis is given by y = –f(x): every y‑coordinate changes sign. Reflection in the y‑axis is given by y = f(–x): every x‑ ay produce the same graph, but in general they are different. For instance, reflecting y = √x in the y‑axis gives y = √–x, which is only defined for x ≤ 0.
关于 x 轴的反射由 y = –f(x) 实现：所有 y 坐标变号。关于 y 轴的反射由 y = f(–x) 实现：所有 x 坐标变号。对于偶函数，这两种反射可能产生相同的图像，但一般来说它们是不同的。例如，将 y = √x 关于 y 轴反射得到 y = √–x，该函数仅在 x ≤ 0 时有定义。
5. Vertical Stretches | 垂直拉伸
A vertical stretch multiplies all y‑ y a factor a. The transformation y = a f(x) stretches the graph vertically by factor a. If |a| > 1, the graph is stretched away from the x‑axis; if 0 < |a| < 1, it is compressed towards the x‑axis. When a is negative, the graph is also reflected in the x‑axis.
垂直拉伸将所有 y 坐标乘以因子 a。变换 y = a f(x) 会将图像垂直拉伸 a 倍。若 |a| > 1，图像远离 x 轴拉伸；若 0 < |a| < 1，则向 x 轴压缩。当 a 为负数时，图像还会同时关于 x 轴反射。
6. Horizontal Stretches | 水平拉伸
Horizontal stretches affect the x‑coordinates. The transformation y = f(bx) compresses the graph horizontally by factor 1/|b|. If |b| > 1, the graph is squeezed towards the y‑axis; if 0 < |b| < 1, it is stretched away from the y‑axis. Pay attention: the factor is the reciprocal of b. For y = f(2x), every x‑coordinate is halved.
水平拉伸影响 x 坐标。变换 y = f(bx) 会将图像水平压缩为原来的 1/|b|。若 |b| > 1，图像向 y 轴挤压；若 0 < |b| < 1，则远离 y 轴拉伸。注意：拉伸因子是 b 的倒数。对于 y = f(2x)，每个 x 坐标都变为原来的一半。
7. Combined Transformations: Order Matters | 组合变换：顺序很重要
When multiple transformations are applied, the order often follows 'horizontal before vertical' ably, work from the inside out. For y = a f(b(x + c)) + d, apply horizontal translation by – h by a, and finally vertical translation by d. Always check with a known point to be sure.
当应用多个变换时，顺序通常遵循“先水平后垂直”原则，更可靠的是由内向外进行处理。对于 y = a f(b(x + c)) + d，首先进行水平平移 –c，然后水平拉伸 1/b，接着垂直拉伸 a，最后垂直平移 d。务必用一个已知点检验以确保正确。
8. Describing Transformations from Graphs | 从图像描述变换
d asked to describe the transformation that maps f(x) to g(x), compare key features such as intercepts, asymptotes, and turning points. Express g(x) in the form a f(b(x + c)) + d and state each transformation in words, specifying direction and magnitude.
若给出 y = g(x) 的图像，要求你描述从 f(x) 到 g(x) 的变换，请比较截距、渐近线和转折点等关键特征。将 g(x) 表示为 a f(b(x + c)) + d 的形式，并用文字逐一说明每个变换，指明方向和大小。
9. Finding the Image of a Point | 求点的像
pply the changes directly to the coordinates. For y = a f(b(x + c)) + d, the image is ( (p – s formula helps check whether your transformed graph passes through the correct locations.
要求原图上特定点 (p, q) 在某变换下的像，可直接对坐标应用变化。对于 y = a f(b(x + c)) + d，像点为 ( (p – c)/b , a q + d )。这一公式有助于检验变换后的图像是否经过正确的位置。
10. Applying Transformations to Specific Functions | 对特定函数应用变换
ncludes quadratic, cubic, exponential, and trigonometric functions. For f(x) = x², y = 2(x – ntal translation 3 right, and vertical translation 1 up. For f(x) = sin x, y = sin(2x + π) – s a horizontal compression by 1/2, a phase shift left π/2, and a vertical shift down by 1.
练习3F常包括二次函数、三次函数、指数函数和三角函数。对于 f(x) = x²，y = 2(x – 3)² + 1 表示垂直拉伸2倍、水平右移3、垂直上移1。对于 f(x) = sin x，y = sin(2x + π) – 1 则包含水平压缩1/2、向左的相位移 π/2 以及向下平移1。
11. Common Mistakes to Avoid | 常见错误避免
Many students confuse the direction of horizontal translations or apply stretches in the wrong order. Do not forget that f(x + 2) shifts left, not right. Also, when multiple horizontal transformations are combined, factor out the coefficient of x first: f(2x + 4) should be written as f(2(x + 2)) to see the correct translation and stretch.
许多学生混淆水平平移的方向，或以错误的顺序进行拉伸。不要忘记 f(x + 2) 是向左平移，而非向右。此外，当组合多个水平变换时，要先提取 x 的系数：f(2x + 4) 应写为 f(2(x + 2))，这样才能看清正确的平移和拉伸。
12. Practice Summary | 练习总结
To excel in Exercise 3F, practise by sketching the original and transformed graphs on the same axes. Label key points and asymptotes before and after each transformation. Work systematically through combined transformations using the inside‑out rule, and always verify with a test point. With consistent practice, graph transformations will become intuitive and precise.
要想在练习3F中取得优秀，要在同一坐标系下勾勒原图和变换后的图像。标记每次变换前后的关键点和渐近线。按照由内向外的规则系统地处理组合变换，并始终用检验点进行验证。通过持续练习，图像变换将变得直观而精确。
Published by TutorHao | Mathematics Revision Series | aleveler.com
更多咨询请联系16621398022（同微信）
或截图后用「扫一扫 → 相册」选择本图